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 wu :: forums    riddles    putnam exam (pure math) (Moderators: Icarus, Eigenray, SMQ, william wu, Grimbal, towr)    Factorial Sum « Previous topic | Next topic »
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ThudnBlunder
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 Factorial Sum   « on: Feb 15th, 2011, 2:14pm » Quote Modify

49
Evaluate if you please (-1)k C(99,2k) = C(99,0) - C(99,2) + C(99,4) - ......... + C(99,96) - C(99,98)
k=0
 « Last Edit: Feb 15th, 2011, 2:17pm by ThudnBlunder » IP Logged

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towr
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 Re: Factorial Sum   « Reply #1 on: Feb 16th, 2011, 9:46am » Quote Modify

[ 99 C(99,k) k=0 ik + 99k=0 C(99,k) (-i)k ]/2
[(1+i)99 + (1-i)99]/2
{[2 epi/4 i]99 + [2 e-pi/4 i]99}/2
297 [e99 pi/4 i + e-99 pi/4 i]
297 [e3 pi/4 i + e-3 pi/4 i]
- 298
- 249
 « Last Edit: Feb 16th, 2011, 9:48am by towr » IP Logged

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ThudnBlunder
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 Re: Factorial Sum   « Reply #2 on: Feb 16th, 2011, 10:31am » Quote Modify

Nicely done, towr.
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